Elliptic Curve Crypto & ECC Diffie-Hellman Presenter: Le Thanh Binh

Outline 1.What is Elliptic Curve ? 2.Addition on an elliptic curve 3.Elliptic Curve Crypto (ECC) 4.ECC Diffie–Hellman

Lets start with a puzzle… What is the number of balls that may be piled as a square pyramid and also rearranged into a square array?

Lets start with a puzzle… What is the number of balls that may be piled as a square pyramid and also rearranged into a square array? Sol: Let x be the height of the pyramid… Thus, We also want this to be a square: Hence, (4900 balls) y=70

Graphical Representation Curves of this nature are called ELLIPTIC CURVES

What is an Elliptic Curve ? An elliptic curve E is the graph of an equation of the form y 2 = x 3 + ax + b “Elliptic curve” is not a cryptosystem

Addition on an elliptic curve mod p The Elliptic Curve E

Addition on an elliptic curve mod p Start with two points P and Q on E

Addition on an elliptic curve mod p Draw the line L through P and Q.

Addition on an elliptic curve mod p The line L intersects the cubic curve E in a third point. Call that third point R.

Addition on an elliptic curve mod p Draw the vertical line through R. It hits E in another point.

Addition on an elliptic curve mod p We define the sum of P and Q on E to be the reflected point. We denote it by P ⊕ Q or just P + Q

Addition on an elliptic curve mod p If P 1 and P 2 are on E, we can define P 3 = P 1 + P 2 P1P1 P2P2 P3P3 x y

Addition on an elliptic curve mod p Suppose that we want to add the points on the elliptic curve Let the line connecting P 1 to P 2 be L : y = mx + v Explicitly, the slope and y-intercept of L are given by

Addition on an elliptic curve mod p We find the intersection of by solving We already know that x1 and x2 are solutions, so we can find the third solution x3 by comparing the two sides of Equating the coefficients and hence Then we compute y3 And finally

Addition on an elliptic curve mod p Consider y 2 = x 3 + 2x + 3 (mod 5) x = 0  y 2 = 3  no solution (mod 5) x = 1  y 2 = 6 = 1  y = 1,4 (mod 5) x = 2  y 2 = 15 = 0  y = 0 (mod 5) x = 3  y 2 = 36 = 1  y = 1,4 (mod 5) x = 4  y 2 = 75 = 0  y = 0 (mod 5) Then points on the elliptic curve are (1,1) (1,4) (2,0) (3,1) (3,4) (4,0) and the point at infinity:  What is (1,4) + (3,1) = P 3 = (x 3,y 3 )?

Consider y 2 = x 3 + 2x + 3 (mod 5) What is (1,4) + (3,1) = P 3 = (x 3,y 3 )? P1P1 P2P2 P3P3 x yy=mx+v y 2 =x 3 +Ax+B Addition on an elliptic curve mod p m= (1-4)  (3-1) -1 = -3  2 -1 ((a mod n)(b mod n)) mod n = ab mod n = (2)(3) mod 5 = 6 mod 5 = 1 = ((-3 mod 5)(2 -1 mod 5))mod5

Consider y 2 = x 3 + 2x + 3 (mod 5) What is (1,4) + (3,1) = P 3 = (x 3,y 3 )? Addition on an elliptic curve mod p m= 1 x 3 = = 2 (mod 5) y 3 = 1(1-2) - 4 = 0 (mod 5) On this curve, (1,4) + (3,1) = (2,0) P1P1 P2P2 P3P3 x yy=mx+v y 2 =x 3 +Ax+B

Elliptic Curve Crypto (ECC) Elliptic curve cryptography [ECC] is a public-key cryptosystem (just like RSA)

Public Private Public-key cryptosystem

Public Public-key cryptosystem

??? Secret Public-key cryptosystem

Addition on an elliptic curve

ECC Diffie-Hellman Public: Elliptic curve and point (x,y) on curve Private: Alice’s A and Bob’s B Alice, A Bob, B A(x,y) B(x,y)  Alice computes A(B(x,y))  Bob computes B(A(x,y))  These are the same since AB = BA

Public: Curve y 2 = x 3 + 7x + b (mod 37) and point (2,5)  b = 3 Alice’s private: A = 4 Bob’s private: B = 7 Alice sends Bob: 4(2,5) = (7,32) Bob sends Alice: 7(2,5) = (18,35) Alice computes: 4(18,35) = (22,1) Bob computes: 7(7,32) = (22,1) ECC Diffie–Hellman - Example Addition 4 times

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